Investor Psychology and Security Market Under- and Overreactions

THE JOURNAL OF FINANCE • VOL. LIII, NO. 6 • DECEMBER 1998

KENT DANIEL, DAVID HIRSHLEIFER, and AVANIDHAR SUBRAHMANYAM*

ABSTRACT We propose a theory of securities market under- and overreactions based on two well-known psychological biases: investor overconfidence about the precision of private information; and biased self-attribution, which causes asymmetric shifts in investors’ confidence as a function of their investment outcomes. We show that overconfidence implies negative long-lag autocorrelations, excess volatility, and, when managerial actions are correlated with stock mispricing, public-event-based return predictability. Biased self-attribution adds positive short-lag autocorrelations ~“momentum”!, short-run earnings “drift,” but negative correlation between future returns and long-term past stock market and accounting performance. The theory also offers several untested implications and implications for corporate financial policy.

IN RECENT YEARS A BODY OF evidence on security returns has presented a sharp challenge to the traditional view that securities are rationally priced to reflect all publicly available information. Some of the more pervasive anomalies can be classified as follows ~Appendix A cites the relevant literature!: 1. Event-based return predictability ~public-event-date average stock returns of the same sign as average subsequent long-run abnormal performance! 2. Short-term momentum ~positive short-term autocorrelation of stock returns, for individual stocks and the market as a whole!

*Daniel is at Northwestern University and NBER, Hirshleifer is at the University of Mich- igan, Ann Arbor, and Subrahmanyam is at the University of California at Los Angeles. We thank two anonymous referees, the editor ~René Stulz!, Michael Brennan, Steve Buser, Werner DeBondt, Eugene Fama, Simon Gervais, Robert Jones, Blake LeBaron, Tim Opler, Canice Pren- dergast, Andrei Shleifer, Matt Spiegel, Siew Hong Teoh, and Sheridan Titman for helpful comments and discussions, Robert Noah for excellent research assistance, and participants in the National Bureau of Economic Research 1996 Asset Pricing Meeting, and 1997 Behavioral Fi- nance Meeting, the 1997 Western Finance Association Meetings, the 1997 University of Chicago Economics of Uncertainty Workshop, and finance workshops at the Securities and Exchange Commission and the following universities: University of California at Berkeley, University of California at Los Angeles, Columbia University, University of Florida, University of Houston, University of Michigan, London Business School, London School of Economics, Northwestern University, Ohio State University, Stanford University, and Washington University at St. Louis for helpful comments. Hirshleifer thanks the Nippon Telephone and Telegraph Program of Asian Finance and Economics for financial support. 1839 1840 The Journal of Finance

3. Long-term reversal ~negative autocorrelation of short-term returns separated by long lags, or “overreaction”! 4. High volatility of asset prices relative to fundamentals 5. Short-run post-earnings announcement stock price “drift” in the direction indicated by the earnings surprise, but abnormal stock price performance in the opposite direction of long-term earnings changes.

There remains disagreement over the interpretation of the above evidence of predictability. One possibility is that these anomalies are chance deviations to be expected under market efficiency ~Fama ~1998!!. We believe the evidence does not accord with this viewpoint because some of the return patterns are strong and regular. The size, book-to-market, and momentum effects are present both internationally and in different time periods. Also, the pattern mentioned in ~1! above obtains for the great majority of event studies. Alternatively, these patterns could represent variations in rational risk premia. However, based on the high Sharpe ratios ~relative to the market! apparently achievable with simple trading strategies ~MacKinlay ~1995!!, any asset pricing model consistent with these patterns would have to have extremely variable marginal utility across states. Campbell and Cochrane ~1994! find that a utility function with extreme habit persistence is required to explain the predictable variation in market returns. To be consistent with cross-sectional predictability findings ~e.g., on size, book-to-market, and momentum!, a model would presumably require even more extreme variation in marginal utilities. Also, the model would require that marginal utilities covary strongly with the returns on the size, book-to-market, and momentum portfolios. But when the data are examined, no such correlation is obvious. Given this evidence, it seems reasonable to consider explanations for the observed return patterns based on imperfect rationality. Moreover, there are important corporate financing and payout patterns that seem potentially related to market anomalies. Firms tend to issue equity ~rather than debt! after rises in market value and when the firm or industry book-to-market ratio is low. There are industry-specific financing and repurchase booms, perhaps designed to exploit industry-level mispric- ings. Transactions such as takeovers that often rely on securities financing are also prone to industry booms and quiet periods. Although it is not obvious how the empirical securities market phenomena can be captured plausibly in a model based on perfect investor rationality, no psychological ~“behavioral”! theory for these phenomena has won general acceptance. Some aspects of the patterns seem contradictory, such as appar- ent market underreaction in some contexts and overreaction in others. Ex- planations have been offered for particular anomalies, but we have lacked both an integrated theory to explain these phenomena and out-of-sample empirical implications to test proposed explanations. A general criticism often raised by economists against psychological theo- ries is that, in a given economic setting, the universe of conceivable irratio- nal behavior patterns is essentially unrestricted. Thus, it is sometimes claimed Investor Psychology and Market Reactions 1841 that allowing for irrationality opens a Pandora’s box of ad hoc stories that will have little out-of-sample predictive power. However, DeBondt and Tha- ler ~1995! argue that a good psychological finance theory will be grounded on psychological evidence about how people actually behave. We concur, and also believe that such a theory should allow for the rational side of investor decisions. To deserve consideration a theory should be parsimonious, explain a range of anomalous patterns in different contexts, and generate new empirical implications. The goal of this paper is to develop such a theory of security markets. Our theory is based on investor overconfidence, and variations in confidence arising from biased self-attribution. The premise of investor overconfidence is derived from a large body of evidence from cognitive psychological experiments and surveys ~summarized in Section 1! which shows that individuals overestimate their own abilities in various contexts. In financial markets, analysts and investors generate information for trading through means, such as interviewing management, verifying rumors, and analyzing financial statements, that can be executed with varying de- grees of skill. If an investor overestimates his ability to generate information, or to identify the significance of existing data that others neglect, he will underestimate his forecast errors. If he is more overconfident about signals or assessments with which he has greater personal involvement, he will tend to be overconfident about the information he has generated but not about public signals. Thus, we define an overconfident investor as one who overestimates the precision of his private information signal, but not of information signals publicly received by all. We find that the overconfident-informed overweight the private signal relative to the prior, causing the stock price to overreact. When noisy public information signals arrive, the inefficient deviation of the price is partially corrected, on average. On subsequent dates, as more public information arrives, the price, on average, moves still closer to the full-information value. Thus, a central theme of this paper is that stock prices overreact to private information signals and underreact to public signals. We show that this overreaction-correction pattern is consistent with long-run negative autocorrelation in stock returns, with unconditional excess volatility ~in excess of what would obtain with fully rational investors!, and with further implications for volatility conditional on the type of signal. The market’s tendency to over- or underreact to different types of information allows us to address the remarkable pattern that the average announcement date returns in virtually all event studies are of the same sign as the average post-event abnormal returns. Suppose that the market observes a public action taken by an informed party such as a firm at least partly in response to market mispricing. For example, a rationally managed firm may tend to buy back more of its stock when managers believe their stock is undervalued by the market. In such cases, the corporate event will reflect the managers’ belief about the market valuation error, and will therefore predict future abnormal returns. In particular, repurchases, reflecting undervaluation, will predict positive abnormal returns, and equity offerings 1842 The Journal of Finance will predict the opposite. More generally, actions taken by any informed party ~such as a manager or analyst! in response to mispricing will predict future returns. Consistent with this implication, many events studied in the empirical literature can reasonably be viewed as being responsive to mispricing, and have the abnormal return pattern discussed above. Section II.B.4 offers several additional implications about the occurrence of and price patterns around corporate events and for corporate policy that are either untested or have been confirmed only on a few specific events. The empirical psychology literature reports not just overconfidence, but that as individuals observe the outcomes of their actions, they update their confidence in their own ability in a biased manner. According to attribution theory ~Bem ~1965!!, individuals too strongly attribute events that confirm the validity of their actions to high ability, and events that disconfirm the action to external noise or sabotage. ~This relates to the notion of cognitive dissonance, in which individuals internally suppress information that con- flicts with past choices.! If an investor trades based on a private signal, we say that a later public signal confirms the trade if it has the same sign ~good news arrives after a buy, or bad news after a sell!. We assume that when an investor receives confirming public information, his confidence rises, but disconfirming information causes confidence to fall only modestly, if at all. Thus, if an individual begins with un- biased beliefs about his ability, new public signals on average are viewed as confirming the validity of his private signal. This suggests that public information can trigger further overreaction to a preceding private signal. We show that such continuing overreaction causes momentum in security prices, but that such momentum is eventually reversed as further public information gradually draws the price back toward fundamentals. Thus, biased self-attribution implies short-run momentum and long-term reversals. The dynamic analysis based on biased self-attribution can also lead to a lag-dependent response to corporate events. Cash flow or earnings surprises at first tend to reinforce confidence, causing a same-direction average stock price trend. Later reversal of overreaction can lead to an opposing stock price trend. Thus, the analysis is consistent with both short term post- announcement stock price trends in the same direction as earnings surprises and later reversals. In our model, investors are quasi-rational in that they are Bayesian opti- mizers except for their overassessment of valid private information, and their biased updating of this precision. A frequent objection to models that explain price anomalies as market inefficiencies is that fully rational investors should be able to profit by trading against the mispricing. If wealth flows from quasi-rational to smart traders, eventually the smart traders may dominate price-setting. However, for several reasons, we do not find this argument to be compelling, as discussed in the conclusion. Several other papers model overconfidence in various contexts. De Long et al. ~1991! examine the profits of traders who underestimate risk when prices are exogenous. Hirshleifer, Subrahmanyam, and Titman ~1994! examine how analyst0traders who overestimate the probability that they receive Investor Psychology and Market Reactions 1843 information before others will tend to herd in selecting stocks to study. Kyle and Wang ~1997!, Odean ~1998!, and Wang ~1998! provide specifications of overconfidence as overestimation of information precision, but do not distin- guish between private and public signals in this regard ~see also Caballé and Sakovics ~1996!!. Odean ~1998! examines overconfidence about, and conse- quent overreaction to, a private signal, which results in excess volatility and negative return autocorrelation. Because our model assumes that investors are overconfident only about private signals, we obtain underreaction as well as overreaction effects. Furthermore, because we consider time-varying confidence, there is continuing overreaction to private signals over time. Thus, in contrast to Odean, we find forces toward positive as well as negative autocorrelation; and we argue that overconfidence can decrease volatility around public news events.1 Daniel, Hirshleifer, and Subrahmanyam ~1998! show that our specifica- tion of overconfidence can help explain several empirical puzzles regarding cross-sectional patterns of security return predictability and investor behavior. These puzzles include the ability of price-based measures ~dividend yield, earnings0price, book-to-market, and firm market value! to predict future stock returns, possible domination of b as a predictor of returns by price- based variables, and differences in the relative ability of different price- based measures to predict returns. A few other recent studies have addressed both overreaction and underreaction in an integrated fashion. Shefrin ~1997! discusses how base rate underweighting can shed light on the anomalous behavior of implied volatilities in options markets. In a contemporaneous paper, Barberis, Shleifer, and Vishny ~1998! offer an explanation for under- and overreactions based on a learning model in which actual earnings follow a random walk, but individuals believe that earnings either follow a steady growth trend or are mean-reverting. Because their focus is on learning about the time-series process of a performance measure such as earnings, they do not address the sporadic events examined in most event studies. In another recent paper, Hong and Stein ~1998! examine a setting where under- and overreactions arise from the interaction of momentum traders and news watchers. Mo- mentum traders make partial use of the information contained in recent price trends and ignore fundamental news. News watchers rationally use fundamental news but ignore prices. Our paper differs in focusing on psychological evidence as a basis for assumptions about investor behavior. The remainder of the paper is structured as follows. Section I describes psychological evidence of overconfidence and self-attribution bias. Section II develops the basic model of overconfidence, and here we describe the economic setting and define overconfidence. We analyze the equilibrium to derive implications about stock price reactions to public versus private news, short-term versus long-term autocorrelations, and volatility. Section III ex-

1 A recent revision of Odean’s paper offers a modified model that allows for underreaction. This is developed in a static setting with no public signals, and therefore does not address issues such as short-term versus long-term return autocorrelations, and event-study anomalies. 1844 The Journal of Finance amines time variation in overconfidence to derive implications about the signs of short-term versus long-term return autocorrelations. Section IV con- cludes by summarizing our findings, relating our analysis to the literature on exogenous noise trading, and discussing issues related to the survival of overconfident traders in financial markets.

I. Overconfidence and Biased Self-Attribution Our theory relies on two psychological regularities: overconfidence and attribution bias. In their summary of the microfoundations of behavioral finance, DeBondt and Thaler ~1995! state that “perhaps the most robust finding in the psychology of judgment is that people are overconfident.” Ev- idence of overconfidence has been found in several contexts. Examples include psychologists, physicians and nurses, engineers, attorneys, negotiators, entrepreneurs, managers, investment bankers, and market professionals such as security analysts and economic forecasters.2 Further, some evidence suggests that experts tend to be more overconfident than relatively inexperi- enced individuals ~Griffin and Tversky ~1992!!. Psychological evidence also indicates that overconfidence is more severe for diffuse tasks ~e.g., making diagnoses of illnesses!, which require judgment, than for mechanical tasks ~e.g., solving arithmetic problems!; and more severe for tasks with delayed feedback as opposed to tasks that provide immediate and conclusive outcome feedback ~see Einhorn ~1980!!. Fundamental valuation of securities ~forecasting long-term cash flows! requires judgment about open-ended issues, and feedback is noisy and deferred. We therefore focus on the implications of overconfidence for financial markets.3 Our theory assumes that investors view themselves as more able to value securities than they actually are, so that they underestimate their forecast error variance. This is consistent with evidence that people overestimate their own abilities, and perceive themselves more favorably than they are viewed by others.4 Several experimental studies find that individuals underestimate their error variance in making predictions, and overweight their own forecasts relative to those of others.5 The second aspect of our theory is biased self-attribution: The confidence of the investor in our model grows when public information is in agreement with his information, but it does not fall commensurately when public information

2 See respectively: Oskamp ~1965!; Christensen-Szalanski and Bushyhead, ~1981!, Bau- mann, Deber, Thompson, ~1991!; ~Kidd ~1970!; Wagenaar and Keren ~1986!; Neale and Bazer- man ~1990!; Cooper, Woo, and Dunkelberg ~1988!; Russo and Schoemaker ~1992!; Vonholstein ~1972!; Ahlers and Lakonishok ~1983!, Elton, Gruber, and Gultekin ~1984!, Froot and Frankel ~1989!, DeBondt and Thaler ~1990!, DeBondt ~1991!. See Odean ~1998! for a good summary of empirical research on overconfidence. 3 Odean ~1998!~Sec. II.D! also makes a good argument for why overconfidence should dominate in financial markets. Also, Bernardo and Welch ~1998! offer an evolutionary explanation for why individuals should be overconfident. 4 Greenwald ~1980!, Svenson ~1981!, Cooper et al. 1988, and Taylor and Brown ~1988!. 5 See Alpert and Raiffa ~1982!, Fischhoff, Slovic, and Lichtenstein ~1977!, Batchelor and Dua ~1992!, and the discussions of Lichtenstein, Fischhoff, and Phillips ~1982! and Yates ~1990!. Investor Psychology and Market Reactions 1845 contradicts his private information. The psychological evidence indicates that people tend to credit themselves for past success, and blame external factors for failure ~Fischhoff ~1982!, Langer and Roth ~1975!, Miller and Ross ~1975!, Taylor and Brown ~1988!!. As Langer and Roth ~1975! put it, “Heads I win, tails it’s chance”; see also the discussion of De Long et al. ~1991!.

II. The Basic Model: Constant Confidence This section develops the model with static confidence. Section III consid- ers time-varying confidence. Each member of a continuous mass of agents is overconfident in the sense that if he receives a signal, he overestimates its precision. We refer to those who receive the signal as the informed, I, and those who do not as the uninformed, U. For tractability, we assume that the informed are risk neutral and that the uninformed are risk averse. Each individual is endowed with a basket containing security shares, and a risk-free numeraire that is a claim to one unit of terminal-period wealth. There are four dates. At date 0, individuals begin with their endowments and identical prior beliefs, and trade solely for optimal risk-transfer pur- poses. At date 1, I receives a common noisy private signal about underlying security value and trades with U.6 At date 2, a noisy public signal arrives, and further trade occurs. At date 3, conclusive public information arrives, the security pays a liquidating dividend, and consumption occurs. All random variables are independent and normally distributed. The risky security generates a terminal value of u, which is assumed to be 2 normally distributed with mean uN and variance su . For most of the paper we set uN ϭ 0 without loss of generality. The private information signal received by I at date 1 is

s1 ϭ u ϩ e, ~1!

2 2 where e;N~0,se !~so the signal precision is 10se !. U correctly assesses the 2 2 error variance, but I underestimates it to be sC ,se . The differing beliefs about the noise variance are common knowledge to all.7 Similarly, the date 2 public signal is

s2 ϭ u ϩ h, ~2!

6 Some previous models with common private signals include Grossman and Stiglitz ~1980!, Admati and Pfleiderer ~1988!, and Hirshleifer et al. ~1994!. If some analysts and investors use the same information sources to assess security values, and interpret them in similar ways, the error terms in their signals will be correlated. For simplicity, we assume this correlation is unity; however, similar results would obtain under imperfect ~but nonzero! correlation in signal noise terms. 7 It is not crucial for the analysis that the uninformed correctly assesses the private signal variance, only that he does not underestimate it as much as the informed does. Also, because U does not possess a signal to be overconfident about, he could alternatively be interpreted as a fully rational trader who trades to exploit market mispricing. Furthermore, most of the results will obtain even if investors are symmetrical both in their overconfidence and their signals. Results similar to those we derive would apply in a setting where identical overconfident individuals receive correlated private signals. 1846 The Journal of Finance

2 2 where the noise term h;N~0,sp ! is independent of u and e. Its variance sp is correctly estimated by all investors. Our simplifying assumption that all private information precedes all public information is not needed for the model’s implications. It is essential that at least some noisy public information arrives after a private signal. The model’s implications stand if, more realistically, additional public information precedes or is contemporaneous with the private signal. The formal role of the uninformed in this paper is minimal because prices are set by the risk-neutral informed traders. The rationale for the assumption of overconfidence is that the investor has a personal attachment to his own signal. This implies some other set of investors who do not receive the same signal. Also, similar results will hold if both groups of investors are risk averse, so that both groups influence price. We have verified this ana- lytically in a simplified version of the model. So long as the uninformed are not risk-neutral price setters, the overconfident informed will push price away from fully rational values in the direction described here.

A. Equilibrium Prices and Trades Because the informed traders are risk neutral, prices at each date satisfy

P1 ϭ EC @u6u ϩ e# ~3!

P2 ϭ EC @u6u ϩ e,u ϩ h#, ~4! where the subscript C denotes the fact that the expectation operator is cal- ϭ u culated based on the informed traders’ confident beliefs. Trivially, P3 .By standard properties of normal variables ~Anderson ~1984!, Chap. 2!,

2 su ϭ ~u ϩ e! ~5! P1 2 2 su ϩ sC

2 2 2 2 2 2 2 su ~sC ϩ sp ! su sp su sC P2 ϭ u ϩ e ϩ h, ~6! D D D

2 2 2 2 2 where D [su~sC ϩ sp ! ϩ sC sp .

B. Implications for Price Behavior This section examines the implications of static confidence for over- and underreactions to information and empirical securities returns patterns. Sub- section B.1 examines price reactions to public and private information, subsection B.2 examines the implications for price-change autocorrelations, and subsection B.3 examines implications for event-studies. Subsection B.4 discusses some as-yet-untested empirical implications of the model. Investor Psychology and Market Reactions 1847

Figure 1. Average price as a function of time with overconfident investors. This figure shows price as a function of time for the dynamic model of Section III with ~dashed line! and without ~solid line! self-attribution bias.

B.1. Overreaction and Underreaction Figure 1 illustrates the average price path following a positive ~upper curve! or negative ~lower curve! date 1 private signal ~date 3' of the graph has not yet been introduced!. At this point we focus on the solid lines. The upper curve, an impulse-response function, shows the expected prices conditional on a private signal of unit magnitude arriving at time 1. The thin horizontal line shows the fully rational price level. Overconfidence in the private signal u ϩ e causes the date 1 stock price to overreact to this new information. At date 2, when noisy public information signals arrive, the inefficient deviation of the price is partially corrected, on average. The same is true on subsequent public information arrival dates. We call the part of the impulse response prior to the peak or trough the overreaction phase, and the later section the correction phase. This overreaction and correction imply that the covariance between the ~ Ϫ Ϫ ! date 1 price change and the date 2 price change, cov P2 P1, P1 P0 ,is negative. ~Appendix B provides detailed expressions for the covariances described here.! Further, the overreaction to the private signal is partially corrected by the date 2 public signal, and fully corrected upon release of the ~ Ϫ Ϫ ! Ͻ date 3 public signal, so that cov P3 P1, P1 P0 0. This price change reversal arises from the continuing correction to the date 1 overreaction. Finally, the continuing correction starting at date 2 and ending at date 3 causes price changes at the time of and subsequent to the public signal to be ~ Ϫ Ϫ ! Ͼ positively correlated, so that cov P3 P2, P2 P1 0. We thus have the following proposition. 1848 The Journal of Finance

PROPOSITION 1: If investors are overconfident, then:

1. Price moves resulting from private information arrival are on average partially reversed in the long run. 2. Price moves in reaction to the arrival of public information are positively correlated with later price changes.

The pattern of correlations described in Proposition 1 is potentially testable by examining whether long-run reversals following days with public news events are smaller than reversals on days without such events. The price behavior around public announcements has implications for corporate event studies ~see Subsection B.3!.

B.2. Unconditional Serial Correlations and Volatility Return autocorrelations in well-known studies of momentum and reversal are calculated without conditioning on the arrival of a public information signal. To calculate a return autocorrelation that does not condition on whether private versus public information has arrived, consider an experiment where the econometrician randomly picks consecutive dates for price changes ~dates 1 and 2, versus dates 2 and 3!. The date 2 and 3 price changes are positively correlated, but the date 1 and 2 price changes are negatively correlated. Suppose that the econometrician is equally likely to pick either pair of consecutive dates. Then the overall autocorrelation is negative.

PROPOSITION 2: If investors are overconfident, price changes are unconditionally negatively autocorrelated at both short and long lags.

Thus, the constant-confidence model accords with long-run reversals ~negative long-lag autocorrelations! but not with short-term momentum ~positive short-lag autocorrelation!. However, the short-lag autocorrelation will be positive in a setting where the extremum in the impulse response function is sufficiently smooth, because the negative autocovariance of price changes surrounding a smooth extremum will be low in absolute terms. Such a setting, based on biased self-attribution and outcome-dependent confidence, is considered in Section III. Overconfidence causes wider swings at date 1 away from fundamentals, ~ ~ Ϫ !! thereby causing excess price volatility around private signals var P1 P0 , as in Odean ~1998!. Greater overconfidence also causes relative underweighting of the public signal, which tends to reduce date 2 variance. However, the wide date 1 swings create a greater need for corrective price moves at dates 2 and 3, so that greater overconfidence can either decrease or increase the ~ ~ Ϫ !! ~ volatility around public signals var P2 P1 . Explicit expressions for the variances of this section are contained in Appendix B.! Consider again an econometrician who does not condition on the occurrence of private or public news arrival. When calculating price change variances he gives equal weight Ϫ Ϫ Ϫ to price changes P1 P0, P2 P1, and P3 P2. The unconditional volatility Investor Psychology and Market Reactions 1849

~ Ϫ ! ~ Ϫ ! is therefore just the arithmetic mean of var P3 P2 , var P2 P1 , and ~ Ϫ ! var P1 P0 . Excess volatility is the difference between the volatility with overconfidence and the volatility when the noise variance is perceived correctly. Let the subscript R denote the volatility if all individuals were rational. We define the date t proportional excess volatility as

var~P Ϫ P Ϫ ! Ϫ var ~P Ϫ P Ϫ ! E t t 1 R t t 1 Vt [ . ~7! varR~Pt Ϫ PtϪ1!

PROPOSITION 3: 1. Overconfidence increases volatility around private signals, can increase or decrease volatility around public signals, and increases unconditional volatility. 2. The proportional excess volatility is greater around the private signal than around the public signal. Thus, consistent with the findings of Odean ~1998!, when there are only private signals, there is a general tendency for overconfidence to create excess volatility. Excess volatility is not an automatic implication of any model with imperfect rationality. For example, if investors are underconfi- 2 2 dent, sC .se, then there will be insufficient volatility relative to the rational level. Also, in contrast to Odean, Proposition 3 implies that in samples broken down by types of news event, either excess or deficient volatility may be possible.

B.3. Event Study Implications Many recent studies have investigated abnormal average return performance or “drift” following public news arrival. As mentioned in the introduction, a striking regularity in virtually all of these studies is that average post-event abnormal price trends are of the same sign as the average initial event-date reaction. We now slightly generalize the model to address this event-based return predictability. Sophisticated managers or analysts who are not overconfident are likely to selectively undertake certain visible actions, such as repurchasing shares or making buy recommendations, when a firm’s shares are undervalued by the market. We will show that the nature of the stock price reaction to an event depends critically on whether or not the event is related to the mispricing by the market. We assume that the date 2 signal is no longer public, but is instead received privately by the firm’s manager ~or other individual such as an analyst!, and that this individual takes an action ~the “event”! that is publicly C observed and fully reveals the signal. Let P2 ~s2! be the valuation that would be placed on the security by an overconfident investor at date 2 were he to ~ observe the signal s2 in addition to his signal s1. Because we examine events ! that fully reveal s2, this is in equilibrium just the post-event stock price P2. 1850 The Journal of Finance

R Let P2 ~s2! be the comparable valuation that would be set by a fully rational R investor. The date 2 mispricing then is defined as the difference P2 ~s2! Ϫ C P2 ~s2!. We define different kinds of events as follows.

Definition. An event is a random variable that depends only on the information signals s1 and s2.Anonselective event is an event that is independent R C of the date 2 mispricing P2 ~s2! Ϫ P2 ~s2!.Aselective event is an event whose occurrence and0or magnitude depends on the date 2 mispricing.

A simple type of nonselective event is a random variable that is linearly related only to the second signal s2.

PROPOSITION 4: If overconfident investors observe a nonselective event: 1. The true expected post-announcement abnormal price change is zero. 2. Conditional on the pre-event return, the covariance between the announcement-date and the post-announcement price change is posi- ~ Ϫ 6 Ϫ ! Ͼ tive; that is, cov P3 P2, s2 P1 P0 0. Because a nonselective event is an action that is unrelated to the pricing error at date 2, it tells us nothing about mean future price movements. Al- though the market underreacts to the event, it is equally likely to be under- reacting downward as upward. Part 1 therefore indicates that there will be no systematic post-announcement drift following events that are unrelated to the prior market mispricing. Thus, Proposition 4 refutes the conventional interpretation of drift as being equivalent to underreaction to new information. The lack of event-based predictive power for future returns is surprising given the positive autocorrelation of event-date and post-event price changes ~Proposition 1!. However, even though the event is unrelated to the prior mispricing, the more underpriced the security, the more positive on average will be the stock price reaction to further news. Thus, a favorable event-date stock price change is associated with a positive future average trend. Clearly, then, even though the event itself does not predict future returns, the market is inefficient. Part 2 of Proposition 4 predicts larger post-event average returns the more the nonselective event ~perhaps a cash flow surprise! and the pre-event stock price runup are in opposition ~e.g., positive pre-event runup and negative ! 8 ~ ! event . Intuitively, holding constant the private signal as reflected in P1 , the higher is the public signal, the more likely that the fundamental u is high, and therefore the bigger the average shortfall of the private signal relative to the fundamental. Thus, a higher public signal is associated with a larger ~more positive! post-event return. Both Parts 1 and 2 of Proposition 4 can be tested using data on specific nonselective events. These are presumably events that are not initiated by an informed party such as a manager with an incentive to take into account

8 We thank an anonymous referee for suggesting that we explore this issue. Investor Psychology and Market Reactions 1851 mispricing. Such events might include news about product demand that em- anates from outside of the company ~e.g., news about competitors’ actions!, or regulatory and legislative outcomes ~e.g., FDA decisions on drugs proposed by a pharmaceutical company!. We now show that selective public events, that is, events that are correlated with pre-event stock mispricing, will forecast future price changes. ~ Consider a manager who observes P1 and therefore infers the private signal ! s1 and receives his own signal s2 at date 2. The manager can undertake a debt0equity exchange offering, and the attractiveness of a larger exchange depends on how high the market price is relative to fundamental value. He can condition the size of the offering on the mispricing at date 2, which he knows precisely, because he knows both s1 and s2. It can easily be shown that in this setting the date 2 pricing error is proportional to the expected e* [ @e6 # error in the private signal, E P1,s2 , where the expectation is again taken with respect to rational beliefs. For tractability, we consider selective events that are linear functions of the date 2 mispricing. When e* Ͻ 0, the manager believes the market has undervalued the firm and that the firm can “profit” by exchanging debt for equity; the more undervalued the firm, the greater the size of the offering. If e* Ͼ 0, an equity- for-debt swap would be preferred instead. It is easy to show that

* * E @P3 Ϫ P26e Ͼ 0# Ͻ 0 Ͻ E @P3 Ϫ P26e Ͻ 0#; ~8! that is, events taken in response to market undervaluation ~e.g., repurchase! are associated with high post-event returns, and events taken in response to overvaluation ~e.g., new issue! are associated with low post-event returns.

PROPOSITION 5: If investors are overconfident, then selective events that are initiated when the stock is undervalued (overvalued) by the market will on average be associated with positive (negative) announcement-date abnormal price changes and will on average be followed by positive (negative) post- announcement abnormal price changes. In Proposition 4 there was underreaction to news arrival but no post-event drift. Here, drift results from the combination of underreaction and event selection based on market mispricing. Thus, the model offers the new empirical implication that the phenomenon of abnormal post-event drift will be concentrated in events that select for market mispricing. Evidence recently brought to our attention supports this implication. Cornett, Mehran, and Tehranian ~1998! find that involuntary issues undertaken by banks to meet capital requirements are not associated with postevent drift, whereas voluntary bank issues are associated with negative post-event abnormal performance. Because involuntary issues are likely to be less selective than voluntary ones, this evidence is consistent with the model. If the announcement of an upcoming initial public offering ~IPO!, like a seasoned equity offering ~SEO! announcement, reflects managers’ “bad news,” then Proposition 5 implies long-run underperformance following IPOs as well. 1852 The Journal of Finance

We have no data on the announcement-date reaction to an upcoming IPO because IPO firms are private prior to the event. However, the consistent findings of negative stock price reactions to seasoned equity issue announcements, and of inferior post-IPO accounting performance ~Jain and Kini ~1994!, Mikkelson, Partch, and Shah ~1997!, Teoh, Wong, and Rao ~1998!, Loughran and Ritter ~1997!!, suggest that an IPO announcement is indeed, on average, bad news.9 If so, the evidence that IPOs internationally exhibit long-run average underperformance for several years after the issue ~Ritter ~1991!, Loughran, Ritter, and Rydqvist ~1994!! is consistent with the model. The event-based return predictability of Proposition 5 is not equivalent to underreaction to corporate events. Underreaction to public signals ~as implied by overconfidence! induces positive autocorrelation of returns at the event date. However, the event realization ~in contrast to the event-date return! does not predict future abnormal returns unless event size0occurrence is correlated with prior market mispricing. We have interpreted the model in terms of firms buying or selling shares to profit from mispricing. An alternative interpretation is that a manager with favorable information ~e* Ͻ 0! would like to signal good news to the market, and chooses an action ~such as a repurchase, dividend, debt for equity swap, or stock split! to reveal his information. With a continuous signal, such behavior typically leads to full revelation, consistent with our assumption that e* is revealed to the market at the event date.10 Whether the model of this section is consistent with the well-known phenomenon of post-earnings announcement “drift” depends on whether earnings announcements are selective events. Earnings reports are favorably selective if managers report higher earnings, ceteris paribus, when the market undervalues their firm. Managers are motivated to do so if they are averse to low levels of short-term stock price or personal reputation.11 Fur- ther, managers have a great deal of discretion over earnings levels both through accounting adjustments ~accruals!, and by shifting the timing of actual cash flows. Accounting adjustments seem to reflect managers’ inside information, as evidenced by the announcement effect of accruals on returns ~distinct from the effect of cash flows!; see Wilson ~1986!. There is extensive evidence that managers use their accounting discretion strategically to achieve their goals, such as meeting loan covenant requirements, winning proxy fights, obtaining earnings-based bonuses, and avoiding taxes; Teoh, Wong, and Rao

9 The initial positive return relative to issue price ~“underpricing”! is not an announcement reaction to the news that an IPO will occur; this news is released earlier. 10 The model’s event study predictions also apply to events undertaken by outsiders who have information about the firm. An example is an analyst’s recommendation to buy or sell shares of the firm. Thus, the analysis is consistent with evidence on stock price drift following analyst buy and sell recommendations mentioned in Appendix A. 11 Either concave utility or risk of dismissal can make managers averse to a low stock price; a rising disutility from low price is a common model assumption ~see, e.g., Harris and Raviv ~1985!!. If managers prefer a high short-term stock price but thereby risk incurring a penalty for overly aggressive reports, then the net benefit from reporting higher earnings may be greater, ceteris paribus, when the stock is more undervalued. Investor Psychology and Market Reactions 1853

~1998! reference about thirty such studies. If managers adjust earnings selectively, Proposition 5 can account for post-earnings drift. The dynamic confidence setting of Section III provides a distinct explanation for post- earnings announcement drift that obtains even if earnings are nonselective. Because the date 1 expected value of e* is perfectly positively correlated ~ ! with P1 they both are linearly increasing functions of s1 , variables such as 0 ~ Ϫ u!N market book or runup P1 are potential measures of mispricing. We have assumed that the size of a selective event depends on the size of the misvaluation; it follows that the size and sign of the selective event varies with the measures of mispricing. We therefore have the following proposition.

PROPOSITION 6: 1. The expected size of a positive (negative) selective event is increasing (decreasing) in measures of the firm’s mispricing. 2. The probability that a positive (negative) selective event will occur increases (decreases) with measures of the firm’s mispricing. We tentatively identify mispricing with variables that contain market price such as market0book ratios. The analysis then predicts that repurchases and other favorable events will tend to occur when market, industry, or firm market0book or price0earnings ratios are low, and that equity issuance and other adverse selective events will tend to occur when such ratios are high. This is consistent with evidence that the frequency of IPOs is positively related to the market0book ratio in the company’s industrial sectors ~Paga- no, Panetta, and Zingales ~1998!!, and that in many countries the value and number of IPOs is positively associated with stock market levels ~Loughran et al. ~1994!, Rees ~1996!, Ljungqvist ~1997!!. The analysis also implies that event-date price changes ~for a given type of event! should be positively correlated with post-announcement returns. This is just underreaction, and follows under the conditions of Proposition 1.12 Also, in the model, because the pre-event price runup maps one-to-one with market mispricing, better pre-event price performance is associated with worse post- event performance ~either including or excluding the event date!. This follows ~ Ϫ Ϫ ! Ͻ ~ Ϫ Ϫ ! Ͻ because cov P3 P2, P1 P0 0 and cov P3 P1, P1 P0 0. Intuitively, mispricing arises from overreaction to private information, firms select events based on mispricing, and this causes post-event returns to be related to pre- event returns. However, the latter implication is not robust to reasonable generalization of the assumptions to allow for the possibility that public information can arrive at date 0 or date 1. Consider, for example, the case of dividend announcements. Firms that have been performing well enough to generate a lot of cash are more likely to boost dividends. Thus, a dividend increase will be associated not only with market undervaluation at date 2 ~unfavorable date 1 private signal!, but

12 Proposition 1 is based on a nonselective news event, namely, the arrival of s2. Even though s2 is private information here, the result is the same because s2 is fully revealed by the corporate action, so that P2 is identical in all states to what it would be if s2 were made public ~ Ϫ Ϫ ! directly. Thus, cov P3 P2, P2 P1 is the same in both cases. 1854 The Journal of Finance also with good past performance ~favorable date 0 or 1 public signal!. In this scenario, while the event-date and post-event mean abnormal returns are both positive, the sign of the pre-event mean return will be ambiguous. We have verified formally that if the event choice ~dividend! increases with both a past ~date 1! public signal and the degree of market undervaluation, then the event may be associated with a positive average runup, a positive average event date return, and a positive average post-event return.13 More generally, whether prior runup ~or other price-related indicators such the fundamental0price ratios! is a measure of mispricing depends on whether the event in question is mainly selective for mispricing, or depends heavily on past fundamental public performance measures ~such as past earnings!. Many events, such as dividends and stock splits, may be selective owing to a signaling motive. But events in which the firm trades against the market, such as exchange offers, repurchases, and new issues, provide an incentive to earn a trading profit. This provides an incentive to be selective above and beyond any signaling motive. Thus, runup and price0fundamental ratios should be better measures of mispricing for such market-exploitation events than for pure signaling events.

B.4. Empirical Implications The model provides the following implications, which are either untested or have been tested only on a subset of possible events: 1. Average post-event returns of the same sign as average event-date returns for selective events, and zero postevent drift for nonselective events 2. A positive correlation between initial event-date stock price reactions and post-event performance for public events 3. A positive correlation between the size of a selective event ~e.g., a repurchase or the announcement of a toehold stake! and post-event return, but no such correlation for nonselective events ~e.g., news disclosed by outside sources, especially if it is macroeconomic or industry-wide, such as news about product demand or input prices, production pro- cesses, and regulatory events! 4. Larger post-event average returns the more the nonselective event and the pre-event stock price runup are in opposition 5. Greater average long-term reversal of price moves occurring on dates when there are no public news events about a firm reported in public news media than price moves occurring on public event dates

13 Fama ~1998! argues that our approach implies that mean pre-event abnormal returns will have the same sign as mean postevent abnormal returns, and that the evidence does not support this implication. As discussed above, event occurrence is likely to depend on past public information, in which case the model implies that average pre-event runup can have either the same or the opposite sign as average postevent abnormal returns. Propositions 4 and 5 provide model implications for event study returns that are robust with respect to pre-event public information arrival. The evidence generally supports these predictions. Investor Psychology and Market Reactions 1855

6. Greater selective event sizes ~e.g., greater repurchases! when mispricing measures ~e.g., price0fundamental ratios or past runup! are high 7. Greater probability of a good-news ~bad news! selective event when the security is more underpriced ~overpriced!. The overconfidence theory has further implications for managerial policy related to implications ~6! and ~7! above. We expect firms to issue securities when they believe their stocks are overvalued. If investors are overconfident, such overvaluation may be measured by recent increases in firm, industry, or aggregate stock market prices, or with high price0fundamental ratios. Conversely, firms should repurchase after rundowns when the market appears to undervalue the firm. Thus, if managers act to exploit mispricing, there will be both general and industry-specific financing and repurchase booms. The theory also suggests that when the market undervalues the firm, there should be a tilt away from dividends toward repurchase. Further, when a stock is underpriced ~perhaps after rundowns or when firm or aggregate market0book ratios are low!, the firm, acting in current shareholders’ inter- ests should, ceteris paribus, favor rights over public issues. Similarly, the firm should tilt toward debt rather than equity issues to avoid diluting current shareholders. Thus, the theory offers a possible solution to what Opler and Titman ~1996! call a major puzzle from the perspective of optimal capital structure theory, that after a rise in market prices, firms tend to issue more equity rather than debt.14 These predictions seem quite intuitive, making it easy to forget that the directions would reverse in alternative models of market mispricing. For example, in a setting where the market always underreacts, firms with high recent runups or low fundamental0price ratios will, ceteris paribus, tend to be undervalued, so that ~inconsistent with the evidence! we would observe repurchases rather than equity issues in such situations.

III. Outcome-Dependent Confidence The implications described so far are based on a fixed confidence level. However, psychological evidence and theory suggest that actions and resulting outcomes affect confidence; events that confirm an individual’s beliefs and actions tend to boost confidence too much, and disconfirming events weaken confidence too little ~see Section I!. Taking into account this psychological pattern leads to implications similar to those in the static section, except that there is also short-run momentum in stock prices and event- based predictability even for nonselective events.

14 However, Jung, Kim, and Stulz ~1996! find that firms often depart from the pecking order ~i.e., the preference of debt over equity! because of agency considerations, and that debt and equity issuers both have negative average abnormal long-run stock returns that are not sta- tistically different from one another. 1856 The Journal of Finance

Consider an informed individual who initially is not overconfident, and who buys or sells a security based on his private information. A public signal confirms his trade if they have the same sign ~“buy” and a positive signal, or “sell” and a negative signal!. We assume that if the later public signal confirms his trade, the individual becomes more confident, and if it disconfirms his confidence decreases by little or remains constant. This implies that, on average, public information can increase confidence, intensifying overreaction. The continuing overreaction leads to positive autocorrelation during the initial overreaction phase. As repeated public information arrival draws the price back toward fundamentals, the initial overreaction is gradually reversed in the long run. The process described above yields a hump-shaped impulse response function for a private signal, as illustrated by the dashed lines in Figure 1. ~The date 001 line overlaps the solid lines showing the impulse response for the static model.! The figure shows two possible date 1 prices, and the paths for expected price conditional on the date 1 move. It can be seen that with outcome-dependent confidence, there are smooth overreaction and correction phases. Pairs of returns drawn from these phases will be positively correlated, whereas the pair that straddles the extremum will be negatively correlated. The overall autocorrelation involving contiguous price changes will be positive if the extremum-straddling negative correlation is sufficiently small. However, price changes that are separated by long lags are likely to straddle the extremum of the impulse-response function, and will therefore exhibit negative autocorrelations. Thus, the pattern of momentum at short lags and reversal at long lags arises naturally from the model. We present two models with dynamic confidence that capture this intuition. The model presented in Section A below is tractable but highly styl- ized. The model presented in Section B allows us to develop more complex implications, but can only be solved by simulation.

A. The Simple Model with Outcome Dependent Confidence We modify the basic model of Section II as follows. We still allow for, but 2 2 no longer require, initial overconfidence, so sC Յ se . For tractability, the ϭ Ϫ public signal is now discrete, with s2 1or 1 released at date 2. We assume that the precision assessed by the investors at date 2 about their earlier private signal depends on the realization of the public signal in the following way. If

sign~u ϩ e! ϭ sign~s2!, ~9! confidence increases, so investors’ assessment of noise variance decreases to 2 2 sC Ϫ k,0ϽkϽsC.If

sign~u ϩ e! Þ sign~s2!, ~10!

2 confidence remains constant, so noise variance is still believed to be sC . Investor Psychology and Market Reactions 1857

The probability of receiving a public signal ϩ1 is denoted by p. For a high value to be a favorable indicator of value, p must tend to increase with u. However, allowing p to vary with u creates intractable nonnormalities. We therefore examine the limiting case where the signal is virtually pure noise, so that p is a constant. ~Appendix C provides a discrete model that derives similar results using an informative public signal.! Given normality of all random variables, the date 1 price is

2 su P ϭ E @u6u ϩ e# ϭ ~u ϩ e!. ~11! 1 C 2 2 su ϩ sC

ϭ ~u ϩ e! Þ ~ ! The date 0 price P0 0, the prior mean. If sign sign s2 , then confidence is constant. Because the public signal is virtually uninformative, the price ~virtually! does not move at date 2. However, if sign~u ϩ e! ϭ sign ~ ! s2 , then the new price is calculated using the new level of the assessed e variance of . This price, denoted by P2C,is

2 su P ϭ ~u ϩ e!. ~12! 2C 2 2 su ϩsCϪk

A.1. Implications of the Simple Model Explicit calculations and expressions for covariances for this subsection are in Appendix D. It can easily be shown that

cov~P2 Ϫ P1, P1 Ϫ P0! Ͼ 0. ~13!

Thus, the model shows that the overreaction phase, not just the correction phase, can contribute positively to short-term momentum. As a result,

cov~P3 Ϫ P1, P1 Ϫ P0! Ͻ 0; ~14!

cov~P3 Ϫ P2, P2 Ϫ P1! Ͻ 0, ~15! because the dates 1 and 2 overreactions must be reversed in the long term. Intuitively, further dates of noisy public information arrival should eventually cause the mispricing to be corrected ~as long as confidence does not explode infinitely!. This process causes positive autocorrelation during the correction phase, just as in the basic model of Section II. To examine this, let us add a date 3' between dates 2 and 3, where a public signal u ϩ h is released. For simplicity, we assume that overconfidence is not affected by the release of the second public signal. As in Section II, h is a zero mean, 2 normally distributed variable with variance sp , and is independent of all other random variables. The price at date 3' when overconfidence is not 1858 The Journal of Finance revised at date 2 is given by equation ~6!. When overconfidence is revised at ' date 2, the price at date 3 , denoted by P3'C , is given by the same expression 2 2 as equation ~6!, except that sC is replaced by sC Ϫ k; that is,

2 2 2 2 2 2 2 su ~sC Ϫ k ϩ sp ! su sp su ~sC Ϫ k! P3'C ϭ u ϩ e ϩ h, ~16! D D D

2 2 2 2 2 where D [su~sC Ϫ k ϩ sp ! ϩ ~sC Ϫ k!sp . With the extra date added to the model, it is easy to show that all of the remaining single-period price-change autocorrelations are negative except ~ Ϫ ' ' Ϫ ! for cov P3 P3 , P3 P2 , which is positive. This can be explained as follows. Date 2 is the extremum of the impulse response function ~the “hump” or “trough” date after which the average correction begins!. By equation ~D1! in Appendix D and the above, the single-period price-change single-lag autocorrelations that fall entirely within either the overreaction phase or within the correction phase are positive, and the single-period price-change single- lag autocorrelation that straddles the extremum is negative.15 Under appropriate parameter assumptions, the negative single-lag autocorrelation surrounding the extremum is arbitrarily close to zero. This occurs if either the extra overreaction or the start of the correction is weak ~or both!. The extra overreaction is small if confidence is boosted only slightly ~k Ͼ 0 small! when an investor’s trade is confirmed by public news. The initial correction is slight if the further noisy public signal is not very in- 2 formative ~sh large!. When parameter values are such that this straddling autocorrelation is not too large, it is outweighed by the positive autocorrelations during the hearts of the overreaction or correction phases. In other words, an econometrician calculating autocorrelations unconditionally would find, in a large sample, a positive single-lag autocorrelation. In contrast, longer-lag pairs of price changes that straddle the extremum of the impulse response function will tend to be opposed, because a price change drawn from the overreaction phase tends to be negatively correlated with a price change drawn from the correction phase. Thus, the theory provides a joint explanation for both short-term momentum and long-term reversals.

PROPOSITION 7: If investor confidence changes because of biased self-attribution, and if overreaction or correction is sufficiently gradual, then stock price changes exhibit unconditional short-lag positive autocorrelation (“momentum”) and long-lag negative autocorrelation (“reversal”).

According to Jegadeesh and Titman ~1993!, their momentum evidence is “... consistent with delayed price reactions to firm-specific information.” Prop- osition 7 offers a very different possible interpretation, namely, that momentum occurs not because the market is slow to react to news but because the

15 ~ Ϫ Ϫ ! Ͼ ~ Ϫ ' ' Ϫ ! Ͼ ~ ' Ϫ Ϫ ! Ͻ Formally, cov P2 P1, P1 P0 0, cov P3 P3 , P3 P2 0, and cov P3 P2, P2 P1 0. Investor Psychology and Market Reactions 1859 market initially overreacts to the news, and later public news triggers further overreaction to the initial private signal. More generally, Proposition 7 refutes the common casual equating of positive versus negative autocorrelations with underreaction versus overreaction to new information. Though negative autocorrelations result from overreaction in the model, positive autocorrelations also result from continuing overreaction ~followed by underreaction in the correction of this error!. Evidence from the psychological literature suggests that individuals tend to be more overconfident in settings where feedback on their information or decisions is slow or inconclusive than where the feedback is clear and rapid ~Einhorn ~1980!!. Thus, mispricing should be stronger in stocks that require more judgment to evaluate, and where the feedback on this judgment is ambiguous in the short run, such as for growth stocks whose value is, on average, more strongly tied to hard-to-value growth options. This conjecture is consistent with recent work by Daniel and Titman ~1998!, which finds that the momentum effect is strong in growth stocks, but is weak or nonex- istent in value stocks. This line of reasoning also suggests that momentum should be stronger for stocks that are difficult to value, such as those with high R&D expenditures or intangible assets.

B. A Dynamic Model of Outcome-Dependent Confidence We now extend this model to an arbitrary number of periods and present numerical simulations. The analysis implies patterns of security price- change autocorrelations consistent with the findings of Section III.A above. It also yields further implications for the correlation between public information announcements ~such as managers’ forecasts or financial reports of sales, cash flows, or earnings! and future price changes.

B.1. The Model We retain the basic structure considered in earlier sections. We assume that the investor has a prior on the precision of his private signal, and uses an updating rule that reflects self-attribution bias. As before, the ~unobserv- 2 able! value of a share of the firm’s stock is u;D N~0,su !. The public noise 2 variance su is common knowledge. At date 1, each informed investor re- I ϭ uD ϩ eI e;I N~ s2 ! ceives a private signal s1 , where 0, e . At dates 2 through T, fE fE ϭ uD ϩ hI hI hI ; N~ s 2 ! a public signal t is released, t t, where t is i.i.d. and t 0, h . s 2 ⌽ The variance of the noise, h , is also common knowledge. Let t be the average of all public signals through time t:

1 t 1 t ⌽t ϭ ( fE t ϭ u ϩ ( hI t. ~17! ~t Ϫ 1! tϭ2 ~t Ϫ 1! tϭ2

⌽ Ϫ The average public signal t is a sufficient statistic for the t 1 public ⌽E ; N~u s20~ Ϫ !! signals, and t , h t 1 . 1860 The Journal of Finance

As before, an informed investor forms expectations about value rationally ~using Bayesian updating! except for his perceptions of his private informa- 2 tion precision. The error variance se is incorrectly perceived by the investor. 2 He estimates se using an ad hoc rule described below. At time 1, the inves- v ϭ 0s2 tor believes that the precision of his signal, C,1 1 C,1 , is greater than the 2 true precision ve ϭ 10se . At every subsequent release of public information the investor updates his estimate of the noise variance. If the new public ~f ! signal t confirms the investor’s private signal s1, and the private signal is not too far away from the public signal, then the investor becomes more confident in his private signal. If the new public signal disconfirms his private signal, the investor revises the estimated precision downward, but not by as much. Thus, the specific updating rule that we implement is:

sign~s1 Ϫ⌽tϪ1!ϭsign~ft Ϫ⌽tϪ1!and 6s1 Ϫ⌽tϪ16Ͻ2s⌽,t v ϭ ~ ϩ T !v ~ ! if Ά then C, t 1 k C, tϪ1 18 otherwise vC, t ϭ ~1 Ϫ kt !vC, tϪ1,

s ⌽ 16 where ⌽,t is the standard deviation of at time t. We impose the re- striction that kT Ͼ kt Ͼ 0. The ratio ~1 ϩ kT !0~1 Ϫ kt ! is an index of the investor’s attribution bias.17

B.2. The Equilibrium Since the investor is risk neutral and the risk-free rate is zero, at each point in time the stock price is the expectation of its terminal value:

Pt ϭ EC @uD 6s1,f2,...,ft#ϭEC@uD6s1,⌽t#. ~19!

2 2 Define vu ϭ 10su and vh ϭ 10sh . The price of the security at time t is given by:

~t Ϫ 1!vh ⌽ ϩ v s E t C, t 1 Pt ϭ EC @uD 6s1,⌽t # ϭ . ~20! vu ϩ vh ϩ vC, t

⌽ ~ Ϫ !v Recall that the precision of t is t 1 h.

B.3. Simulation Results and Empirical Implications

T 2 2 For the simulation we use the parameters k ϭ 0.75, kt ϭ 0.1, su ϭ se ϭ 1, 2 and sh ϭ 7.5. We also make the investor’s initial estimate of his precision equal to the true precision of his private signal. We perform this simulation

16 For tractability, we assume that the investor forms beliefs as if, at each point in time, he v knows his exact signal precision. Rationally he should allow for the fact that C,t is an estimate. We expect that the essential results are not sensitive to this simplification. 17 Several alternative ad hoc updating rules consistent with this intuition all lead to roughly equivalent results. Investor Psychology and Market Reactions 1861

Period

Figure 2. Average price path following private information shock. This figure shows the average price path calculated using the simulation in Section III.B.3, following a private infor- ϭ ~ ! mation shock s1 1. The price path is shown for the dynamic model with solid line and without ~dashed line! self-attribution bias.

u ϭ u ϩ e 50,000 times, each time redrawing the value , the private signal s1 , f ϭ and the public information set t,fort 2,...T. It is useful to first illustrate the dynamic price path implied by the model u for specific realizations of s1 and . Figure 2 shows the average price path ϭ u ϭ following a private signal of s1 1 when 0, so that the informed investors’ signal is unduly favorable. The price initially jumps from 0 up to 0.5, a rational assessment. On average, the price continues moving up, reaching a maximum of 0.7366 in period 16. The average price then declines, and eventually asymptotes to zero. Thus, there is an initial overreaction phase in which the price moves away from the true value as the investor’s attribution bias causes him to place more weight, on average, on his private information. Eventually the public information become precise enough that the investor revises his valuation of the security downward. This is the correction phase. A similar hump-shaped pattern holds for an investors’ self-perceived precision ~confidence! as a function of time. This changing confidence is the source of the overreacting average price trend. Figure 3 presents the unconditional average autocorrelations ~at lags be- ! uD I tween 1 period and 119 periods , where and s1 are now resampled for each iteration. This figure confirms the intuition derived from Figure 2 that short- lag price change autocorrelations should be positive and long-lag autocorrelations should be negative. 1862 The Journal of Finance

Lag

Figure 3. Average price change autocorrelations. This figure presents the unconditional average autocorrelations ~at lags between 1 period and 119 periods!, calculated using the simulation described in Section III.B.3.

Several papers examine “long-horizon” regressions of long-period returns on past returns ~see, e.g., Fama and French ~1988!! rather than long-lag autocorrelations of short-period returns. In our model, it is straightforward to show that there is a one-to-one mapping between price change autocorrelations and more standard test statistics such as variance ratios or long- horizon regression coefficients. In unreported simulations, these coefficients exhibit behavior similar to that of the autocorrelations. Short-horizon regression coefficients are positive and long-horizon ones are negative, consistent with empirical literature on momentum and reversals. The conclusions of this simulation are summarized as follows.

Result 1: In the biased self-attribution setting of Section III.B, if the true u ϭ ϭ share value 0 and the initial private signal s1 1, then with sufficient attribution bias the average price at first rises and then gradually declines. This contrasts with a steadily declining price path if there is no attribution bias. In the biased self-attribution setting, average self-perceived precision also initially rises and then declines.

Result 2: In the biased self-attribution setting of Section III.B, short-lag autocorrelations ~correlating single-period price changes with single-period price changes! are positive and long-lag autocorrelations are negative. Investor Psychology and Market Reactions 1863

Result 3: In the biased self-attribution setting of Section III.B, short-term autocorrelations are positive and long-horizon autocorrelations are negative.

Recent research indicates strong and consistent evidence of momentum in the United States and in European countries, but weak and insignificant evidence of momentum in Japan ~see, e.g., Haugen and Baker ~1996! and Daniel, Titman, and Wei ~1996!!. There is corresponding evidence of a difference in biased self-attributions in Western versus Asian groups, especially Japan. For example, Kitayama, Takagi, and Matsumoto ~1995! review twenty-three studies conducted in Japan which find essentially no evidence of self-enhancing biases in attribution. These findings suggest the more general prediction that cultures in which there is little or no self-enhancing attribution bias ~e.g., other Asian countries such as Korea, PRC, and Tai- wan; see the references in Kitayama et al. ~1995!! should have weak momentum effects. De Long et al. ~1990a! have derived security return autocorrelations in a model with mechanistic positive feedback traders. Our approach differs in explicitly modeling the decisions of quasi-rational individuals. Our model provides one possible psychological foundation for a stochastic tendency for trades to be correlated with past price movements, which can create an ap- pearance of positive feedback trading.

B.4. Correlation of Accounting Performance with Subsequent Price Changes Finally, we consider the implications of this model for the correlation between accounting performance and future price changes. Accounting information ~sales, earnings, etc.! can be thought of as noisy public signals about u, so in this subsection we interpret the fs as accounting performance change measures. Consider the first public signal ~at t ϭ 2!. If this is positive, the first private signal was probably also positive. Based on the momentum results in this section, this suggests that prices will continue to increase after the arrival date of the public signal, consistent with empirical evidence on earnings-based return predictability. Eventually prices will decline as the cumulative public signal becomes more precise and informed investors put less weight on their signal. Thus, the analysis of this section suggests that earnings-based return predictability, like stock-price momentum, may be a phenomenon of continuing overreaction.18 In the long run, of course, the security price will return to its full-information value, implying long-run negative correlations between accounting performance and future price changes. This conjecture is consistent with the empirical evidence discussed in Appendix A, though from an empirical standpoint statistical power to detect long-lag autocorrelations is limited.

18 The discussion of event-study implications in Section III.B.3 describes conditions under which post-earnings announcement drift could be an underreaction effect. 1864 The Journal of Finance

Lag

Figure 4. Correlation between information changes and future price changes. This ⌬ t figure shows the set of average sample correlations between the et and price changes periods ⌬ ϭ Ϫ in the future Ptϩt Ptϩt PtϩtϪ1. These are calculated using the simulated dynamic model of Section III.B.3.

To evaluate the above conjecture, we again calculate average correlations fE ~ ϭ ! using our simulation as follows. For each t for t 2,120 we calculate the “earnings” surprise, defined as

⌬et ϭ fE t Ϫ⌽t ϭfEt ϪE@fEt6f2,f3,...,ftϪ1#, ~21!

f the deviation of t from its expected value based on all past public signals. ⌬ Then, we calculate the set of sample correlations between the et and price t ⌬ ϭ Ϫ changes periods in the future Ptϩt Ptϩt PtϩtϪ1. These correlations are then averaged over the Monte Carlo draws. The average correlations are plotted in Figure 4. This simulation yields the following result.

Result 4: In the biased self-attribution setting of Section III.B, short-lag correlations between single-period stock price changes and past earnings are positive, and long-lag correlations can be positive or negative.

To summarize, the analysis suggests that the conclusion from the basic model that investors overreact to private signals holds in the dynamic model. Though investors underreact on average to public signals, public signals initially tend to stimulate additional overreaction to a previous private signal. Thus, underreaction is mixed with continuing overreaction. Investor Psychology and Market Reactions 1865

In the model of this section, earnings-based return predictability and momentum both arise from self-attribution bias. Further, the literature cited in Section III.B.3 suggests that the magnitude of this bias varies systemati- cally across countries. Based on these observations, the self-attribution model suggests a positive relationship across international markets between the strength of the momentum effect and that of the post-earnings announcement drift.

IV. Conclusion Empirical securities markets research in the last three decades has presented a body of evidence with systematic patterns that are not easy to explain with rational asset pricing models. Some studies conclude that the market underreacts to information, others find evidence of overreaction. We have lacked a theory to integrate this evidence, and to make predictions about when over- or underreaction will occur. This paper develops a theory based on investor overconfidence and on changes in confidence resulting from biased self-attribution of investment outcomes. The theory implies that investors overreact to private information signals and underreact to public information signals. In contrast with the common correspondence of positive ~negative! return autocorrelations with underreaction ~overreaction! to new information, we show that positive return autocorrelations can be a result of continuing overreaction. This is followed by long-run correction. Thus, short-run positive autocorrelations can be consistent with long-run negative autocorrelations. The theory also offers an explanation for the phenomenon of average public event stock price reactions of the same sign as post-event long-run abnormal returns. This pattern has sometimes been interpreted as market underreaction to the event. We show that underreaction to new public information is neither a necessary nor a sufficient condition for such event- based predictability. Such predictability can arise from underreaction only if the event is chosen in response to market mispricing. Alternatively, predictability can arise when the public event triggers a continuing overreaction. For example, post-earnings announcement drift may be a continuing overreaction triggered by the earnings announcement to pre-event information. The basic noise trading approach to securities markets ~e.g., Grossman and Stiglitz ~1980!, Shiller ~1984!, Kyle ~1985!, Glosten and Milgrom ~1985!, Black ~1986!, De Long et al. ~1990b!, and Campbell and Kyle ~1993!! posits that there is variability in prices arising from unpredictable trading that seems unrelated to valid information. Our approach is based on the premise that an important class of mistakes by investors involves the misinterpre- tation of genuine new private information. Thus, our model endogenously generates trading mistakes that are correlated with fundamentals. Modeling the decision problems of quasi-rational traders imposes restrictions on trade distributions which are not obvious if distributions are imposed exogenously. This structure provides predictions about the dynamic behavior of asset prices 1866 The Journal of Finance which depend on the particular cognitive error that is assumed. For example, underconfidence also gives rise to quasi-rational trading that is correlated with fundamentals, but leads to empirical predictions that are the reverse of what the empirical literature finds. Specifically, if informed in- 2 2 vestors are underconfident ~sC .se !, there is insufficient volatility relative to the rational level, long-run return continuation, and negative correlation between selective events such as repurchase and post-event returns. Of course, one could arbitrarily specify whatever pattern of correlated noise is needed to match empirically observed ex post price patterns. Such an exercise would merely be a relabeling of the puzzle, not a theory. Instead, we examine a form of irrationality consistent with well-documented psychological biases, and our key contribution is to show that these biases induce several of the anomalous price patterns documented in the empirical literature. Some models of exogenous noise trades ~e.g., De Long et al. ~1990b!, Camp- bell and Kyle ~1993!! also imply long-run reversals and excess volatility because of the time-varying risk premia induced by these trades. Our approach additionally reconciles long-run reversals with short-term momentum, ex- plains event-based return predictability, and offers several other distinct empirical predictions ~see Sections II.B.1 through II.B.3!. As noted in the introduction, a possible objection to models with imperfectly rational traders is that wealth may shift from foolish to rational traders until price setting is dominated by rational traders. For example, in our model the overconfident informed traders lose money on average. This outcome is similar to the standard result that informed investors cannot profit from trading with uninformed investors unless there is some “noise” or “supply shock.” However, recent literature has shown that in the long run rational traders may not predominate. De Long et al. ~1990b, 1991! point out that if traders are risk averse, a trader who underestimates risk will allocate more wealth to risky, high expected return assets. If risk-averse traders are overconfident about genuine information signals ~as in our model!, overconfidence allows them to exploit information more effectively. Thus, the expected profits of the overconfident can be greater than those of the fully rational ~see Daniel et al. ~1998!!. Furthermore, owing to biased self-attribution, those who acquire wealth through successful investment may become more overconfident ~see also Ger- vais and Odean ~1998!!. Another distinct benefit of overconfidence is that it can act like a commitment to trade aggressively. Because this may intimi- date competing informed traders, those known to be overconfident may earn higher returns ~see Kyle and Wang ~1997! and Benos ~1998!!. Recent evidence suggests that event-based return predictability varies across stocks ~e.g., Brav and Gompers ~1997!!. Moving beyond the confines of the formal model, we expect the effects of overconfidence to be more severe in less liquid securities and assets. Suppose that all investors are risk averse and that prices are not fully revealing ~perhaps because of noisy liquidity trading!. If rational arbitrageurs face fixed setup costs of learning about a stock, then large liquid stocks will tend to be better arbitraged ~more ratio- Investor Psychology and Market Reactions 1867 nally priced! than small stocks, because it is easier to cover the fixed inves- tigation cost in large, liquid stocks. This suggests greater inefficiencies for small stocks than for large stocks, and for less liquid securities and assets such as real estate than for stocks. Furthermore, because the model is based on overconfidence about private information, the model predicts that return predictability will be strongest in firms with the greatest information asym- metries. This also implies greater inefficiencies in the stock prices of small companies. Furthermore, proxies for information asymmetry such as the adverse selection component of the bid-ask spread should also be positively related to momentum, reversal, and post-event drift. It is an open question whether the overconfident traders in the model can be identified with a specific category of investor, such as institutions, other investment professionals, small individual investors, or all three. Even small individual investors, who presumably have less information, may still be overconfident. The uninformed investors of the model could be interpreted as being contrarian-strategy investors ~whether institutions or individuals!. ~Some smart contrarian investors could be viewed as rational and informed, and including such traders would not change the qualitative nature of the model predictions.! An identification of the confidence characteristics of different observable investor categories may generate further empirical implications, and is an avenue for further research.

Appendix A: Securities Price Patterns This appendix cites the relevant literature for the anomalies mentioned in the first paragraph of the introduction. Out-of-sample tests ~in time and location! have established several of these patterns as regularities.

Underreaction to Public News Events (event-date average stock returns of the same sign as average subsequent long-run abnormal performance) Events for which this has been found include: 1. Stock splits ~Grinblatt, Masulis, and Titman ~1984!, Desai and Jain ~1997!, and Ikenberry, Rankine, and Stice ~1996!! 2. Tender offer and open market repurchases ~Lakonishok and Vermaelen ~1990!, Ikenberry, Lakonishok, and Vermaelen ~1995!! 3. Analyst recommendations ~Groth et al. ~1979!, Bjerring, Lakonishok, and Vermaelen ~1983!, Elton et al. ~1984!, Womack ~1996!, and Michaely and Womack ~1996!!, 4. Dividend initiations and omissions ~Michaely, Womack, and Thaler ~1995!! 5. Seasoned issues of common stock ~Loughran and Ritter ~1995!, Spiess and Affleck-Graves ~1995!, Teoh, Welch, and Wong ~1998!, but see the differing evidence for Japan of Kang, Kim, and Stulz ~1996!! 6. Earnings surprises ~at least for a period after the event!~Bernard and Thomas 1989, 1990!, Brown and Pope ~1996!! 1868 The Journal of Finance

7. Public announcement of previous insider trades ~Seyhun ~1997!; see also Seyhun ~1986, 1987! and Rozeff and Zaman ~1988!! 8. Venture capital share distributions ~Gompers and Lerner ~1997!!. There is also evidence that earnings forecasts underreact to public news, such as quarterly earnings announcements ~Abarbanell ~1991!, Abarbanell and Bernard ~1992!, Mendenhall ~1991!!. An event inconsistent with this generalization is exchange listing ~McConnell and Sanger ~1987!, Dharan and Ikenberry ~1995!!. Fama ~1998! argues that some of these anomalous return patterns are sensitive to empirical methodology. On the other hand, Loughran and Ritter ~1998! argue that the methodology favored by Fama minimizes the power to detect possible misvaluation effects.

Short-Term Momentum (positive short-term autocorrelation of stock returns, for individual stocks and the market as a whole) See Jegadeesh and Titman ~1993! and Daniel ~1996!. “Short” here refers to periods on the order of six to twelve months. At very short horizons there is negative autocorrelation in individual stock returns ~Jegadeesh ~1990!, Le- hmann ~1990!!, probably resulting from bid-ask spreads and other measure- ment problems ~Kaul and Nimalendran ~1990!!. Rouwenhorst ~1998a! finds evidence of momentum in twelve European countries. The effect is stronger for smaller firms. However, Haugen and Baker ~1996! and Daniel ~1996! show that, although there is evidence of a strong book-to-market effect in Japan, there is little or no evidence of a momentum effect. Rouwenhorst ~1998a! reports a strong momentum effect within and across twelve European countries, and Rouwenhorst ~1998b! finds evidence that momentum, firm size, and value predict common stock returns in twenty emerging markets.

Long-Term Reversal (negative autocorrelation of short-term returns separated by long lags, or “overreaction”) Cross-sectionally, see DeBondt and Thaler ~1985, 1987!, and Chopra, La- konishok, and Ritter ~1992!; on robustness issues, see Fama and French ~1996! and Ball, Kothari, and Shanken ~1995!. For the aggregate market, see Fama and French ~1988! and Poterba and Summers ~1988!; internationally, see Richards ~1997!. On the robustness of the finding in the post-WWII period, see Kim, Nelson, and Startz ~1988!, Carmel and Young ~1997!, As- ness ~1995!, and Daniel ~1996!; the latter two papers show that in post- WWII U.S. data, significant cross-sectional ~Asness! and aggregate ~Daniel! long-horizon negative autocorrelations are partly masked by a momentum effect ~positive serial correlation! at approximately a one-year horizon.

Unconditional Excess Volatility of Asset Prices Relative to Fundamentals See Shiller ~1981, 1989!; for critical assessments of this conclusion, see Kleidon ~1986! and Marsh and Merton ~1986!. Investor Psychology and Market Reactions 1869

Abnormal Stock Price Performance in the Opposite Direction of Long-Term Earnings Changes DeBondt and Thaler ~1987!, and Lakonishok, Shleifer, and Vishny ~1994! find a negative relation between long-horizon returns and past financial performance measures such as earnings or sales growth; see however DeChow and Sloan ~1997!. This implies that one or more short-horizon, long-lag regression coefficients must be negative ~proof available on request!. In contrast, Chan, Jegadeesh, and Lakonishok ~1996! do not reject the null of no such a negative relation, perhaps owing to a lack of power in detecting long- run reversals. Also, La Porta et al. ~1997! find large positive returns for value stocks on earnings announcement dates ~and negative for growth stocks!.

Appendix B: Covariance and Variance Calculations for the Basic Model Covariances and Variances of Section II.B

2 2 ~All signs are under the overconfidence assumption that se .sC.! From equation ~6!, the covariance between the date 3 and the date 2 price changes is

6 2 2 2 2 su sC sp ~se Ϫ sC ! ~ Ϫ P , P Ϫ P ! ϭ . ~B1! cov P3 2 2 1 2 2 2 2 2 2 2 2 ~su ϩ sC !@su ~sC ϩ sp ! ϩ sC sp #

2 2 This is positive because se .sC. The covariance between the date 1 price change and the date 2 price change is

6 2 2 2 su sC~se Ϫ sC ! cov~P Ϫ P , P Ϫ P ! ϭϪ , ~B2! 2 1 1 0 2 2 2 2 2 2 2 2 ~su ϩ sC ! @su ~sC ϩ sp ! ϩ sC sp # which, with overconfidence, is negative. The average of the two preceding covariances is given by

4 8 2 2 sC su ~sC Ϫ se ! , ~B3! 2 2 2 2 2 2 2 2 2~su ϩ sC !@sC~su ϩ sp ! ϩ su sp # and is also negative. It is also easy to show that

4 2 2 su~se Ϫ sC ! cov~P Ϫ P , P Ϫ u!N ϭϪ Ͻ 0, ~B4! 3 1 1 2 2 2 ~su ϩ sC !

4 2 2 2 su sp~se Ϫ sC ! ~ Ϫ P , P Ϫ u!N ϭϪ Ͻ 0. ~B5! cov P3 2 1 2 2 2 2 2 2 2 ~su ϩ sC !@sc ~su ϩ sp ! ϩ su sp # 1870 The Journal of Finance

ϭ u ~ ! Because P3 , using the expression for P2 in equation 6 , we have

2 2 2 2 2 2 2 se sh su ~su ϩ sp !~sC Ϫ se ! cov~P Ϫ P ,e * ! ϭ , ~B6! 3 2 2 2 2 2 2 2 2 2 2 2 @se ~su ϩ sp ! ϩ su sp #@sC~su ϩ sp ! ϩ su sp #

2 2 which is positive as long as sC ,se .

Proof of Proposition 3: The variance of the date 2 price change is

4 4 6 8 2 4 2 2 2 2 sC @sC su ϩ su se ϩ su sp ~sC ϩ su ! # var~P Ϫ P ! ϭ , ~B7! 2 1 2 2 2 2 2 2 2 2 2 @sC~sp ϩ su ! ϩ su sp # ~su ϩ sC !

2 which can either increase or decrease in sC . The date 1 price volatility,

4 2 2 su ~su ϩ se ! var~P Ϫ P ! ϭ , ~B8! 1 0 2 2 2 ~su ϩ sC !

2 decreases with sC . ~ Ϫ ! The unconditional volatility is just the average of var P3 P2 , ~ Ϫ ! ~ Ϫ ! var P2 P1 , and var P1 P0 ,

2 4 4 4 2 2 2 2 4 4 6 8 2 4 2 2 2 2 1 se s su ϩ s sh su ~sh ϩ su ! s $s su ϩ su se ϩ su s ~s ϩ su ! % ͫ p C ϩ C C p C 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 @sC~sp ϩ su ! ϩ sp su # ~sC ϩ su ! @sC~sp ϩ su ! ϩ su sp #

4 2 2 su ~su ϩ se ! ϩ ͬ. ~B9! 2 2 2 ~su ϩ sC !

2 2 2 When there is no overconfidence, sC ϭ se , which reduces to su 03. The 2 excess volatility is the difference between the above expression and su 03, 2 2 which is positive so long as there is overconfidence, sC ,se. ~ Ϫ ! ~ Ϫ ! We now calculate var P1 P0 and var P2 P1 when informed agents are rational ~subscripted by R!:

4 su var ~P Ϫ P ! ϭ ~B10! R 1 0 2 2 su ϩ se and

4 4 se su var ~P Ϫ P ! ϭ . ~B11! R 2 1 2 2 2 2 2 2 2 ~se ϩ su !@se ~sp ϩ su ! ϩ sp su # Investor Psychology and Market Reactions 1871

E E Using the calculated variances, V1 Ϫ V2 is equal to the ratio of

2 2 2 2 6 2 2 2 2 2 2 2 ~se ϩ su !~se Ϫ sC !$sC~sp ϩ su !@se ~sp ϩ su ! ϩ sp su #

4 2 2 4 2 2 2 2 2 4 4 ϩ sC @~sp ϩ su !~se ~sp ϩ su ! ϩ 3se sp su ! ϩ 2sp su #

2 2 2 4 2 2 2 2 2 2 2 4 ϩ sC sp su @2se ~sp ϩ su ! ϩ se su ~3sp ϩ 2su ! ϩ sp su #

2 4 4 2 2 ϩ se sp su ~se ϩ su !% ~B12!

4 2 2 2 2 2 2 2 2 2 and se ~sC ϩ su ! @sC~sp ϩ su ! ϩ sp su ! #, and is therefore positive under 2 2 overconfidence ~se .se!. Thus, the proportional difference between overconfident and rational volatilities is greater at date 1 than at date 2.

Proofs of Some Claims in Section B.3

Proof of Part 1 of Proposition 4: Denote the date 2 mispricing as M2. R~ ! C~ ! ϭ R Ϫ C ϭ Suppressing arguments on P2 s2 and P2 s2 , we have that M2 P2 P2 C ϪE @u Ϫ P2 ~s2!6s1,s2#. By the properties of normal random variables, this ϭ u Ϫ C ϩ implies that the variable x P2 M2, which is the residual from the u Ϫ C regression of P2 on s1 and s2, is orthogonal to s1 and s2. Suppose we pick ϭ ~ ! a variable y f s1,s2 which is orthogonal to M2. Such a variable will be ~u Ϫ C ϩ ! ϭ orthogonal to x, so that we have cov P2 M2,y 0. By construction, ~ ! ϭ cov M2, y 0, so it follows from the linearity of the covariance operator C that cov~u Ϫ P2 ,y! ϭ 0. A converse argument shows that if we pick a vari- ' ϭ ~ ! u Ϫ C able y g s1,s2 which is orthogonal to the post-event return P2 then ~ '! ϭ cov M2, y 0. Thus, all functions of s1 and s2 are orthogonal to M2 if and C only if they are orthogonal to the post-event return u Ϫ P2 .

For the specific case when the event depends linearly on s2, by equation ~6!,

2 2 2 2 2 2 sC sp u Ϫ sp su e Ϫ sC su h P Ϫ P ϭ . ~B13! 3 2 2 2 2 2 2 su ~sC ϩ sp ! ϩ sC sp

[uϩh Because s2 , from the above expression it immediately follows that ~ Ϫ ! ϭ cov P3 P2,s2 0, thus showing that events that depend only on s2 are nonselective. Proof of Part 2 of Proposition 4: By standard results for calculating conditional variances of normal variables ~Anderson ~1984!!,

cov~P3 Ϫ P2,s26s1! ϭ cov~P3 Ϫ P2,s26P1 Ϫ P0!

2 4 2 2 sp su ~se Ϫ sC ! ϭ , ~B14! 2 2 2 2 2 2 2 @sC~sp ϩ su ! ϩ sp su #~se ϩsu !

2 2 which is positive under overconfidence ~sC .se !. 1872 The Journal of Finance

Proof of Proposition 5: Using standard normal distribution properties,

2 2 2 2 2 se ~su ϩ sp !~u ϩ e! Ϫsu se ~u ϩ h! e * ϭ E @e6P ,u ϩ h# ϭ . ~B15! 1 2 2 2 2 2 se ~su ϩ sp ! ϩ su sp

It is straightforward to show that the ratio of the date 2 mispricing to e* is

2 2 2 2 2 2 se @sC~su ϩ sp ! ϩ su sp # , ~B16! 2 2 2 2 sp su ~sC Ϫ su ! which is a constant ~for a given level of confidence!. Thus, selective events can alternatively be viewed as events that are linearly related to e*. High values of e* signify overpricing and low values underpricing. The proposition follows by observing that

2 2 2 2 2 2 2 se sh su ~su ϩ sp !~sC Ϫ se ! cov~P Ϫ P ,e * ! ϭ Ͻ 0 ~B17! 3 2 2 2 2 2 2 2 2 2 2 2 @se ~su ϩ sp ! ϩ su sp #@sC~su ϩ sp ! ϩ su sp # and

2 2 4 2 2 2 2 2 sCse su @se ~sp ϩ su ! ϩ sp su # cov~P Ϫ P ,e * ! ϭϪ Ͻ 0. ~B18! 2 1 2 2 2 2 2 2 2 2 ~sC ϩ su !@sC~sp ϩ su ! ϩ sp su #

By the conditioning properties of mean-zero normal distributions, ~ Ϫ e* ! Ͻ @ Ϫ 6e* # cov P3 P2, 0; therefore E P3 P2 can be written in the e* Ͻ @ Ϫ 6e*# Ͻ form k , where k 0 is a constant. Thus, E P3 P2 0ifand only if e* Ͼ 0. This holds for each positive realization of e*, and so @ Ϫ 6e* Ͼ # Ͻ @ Ϫ 6e* Ͻ # Ͼ E P3 P2 0 0. By symmetric reasoning, E P3 P2 0 0. The result for event-date price reactions uses a similar method. Because ~ Ϫ e*! Ͻ @ Ϫ 6e*# Ͻ e* Ͼ cov P2 P1, 0, it follows that E P2 P1 0 if and only if 0.

Proof of Proposition 6: We interpret the “fundamental0price” ratio or “run- uN Ϫ up” as P1. For part 1,

2 2 2 2 2 2 se @se ~sp ϩ su ! ϩ sp su # cov~uN Ϫ P ,e * ! ϭ Ͼ 0. ~B19! 1 2 2 2 2 2 sC~sp ϩ su ! ϩ sp su

By our assumption that the selective event is linearly related to e*, the selective event is positively correlated with the mispricing measure, proving part 1. Investor Psychology and Market Reactions 1873

e* ϭ ϩ For Part 2, note that k1 s1 k2 s2, where

2 2 2 se ~su ϩ sp ! ϭ , ~B20! k1 2 2 2 2 2 sC~sp ϩ su ! ϩ sp su

2 2 se su ϭϪ . ~B21! k2 2 2 2 2 2 sC~sp ϩ su ! ϩ sp su

This implies that the distribution of e* conditional on u ϩ e is normal with mean

2 2 ~k1 ϩ k2!su ϩ k1se ~u ϩ e! ~B22! 2 2 su ϩ se and variance

2 2 2 @~k1 ϩ k2 !su ϩ k1se # ~u ϩ e!. ~B23! 2 2 su ϩ se

The complement of the standardized cumulative normal distribution function of a normal random variable with nonzero mean and variance is increasing in its mean. Because E @e*6u ϩ e# is proportional to u ϩ e, the e* ~ probability conditional on P1 that exceeds a given threshold value indi- cating occurrence of the positive event! is increasing in u ϩ e. The reverse holds for a negative event, proving part ~2!.

Appendix C: Discrete Model of Outcome-Dependent Overconfidence At time 0, u has a value of ϩ1orϪ1 and an expected value of zero. At time

1, the player receives a signal s1, and, at time 2, a signal s2. s1 may be either H or L, and s2 may be either U or D. After each signal, the player updates his prior expected value of u.

Pr~s1 ϭ H6u ϭϩ1!ϭpϭPr~s1 ϭ L6u ϭϪ1!, ~C1!

Pr~s2 ϭ U6u ϭϩ1!ϭqϭPr~s2 ϭ D6u ϭϪ1!. ~C2!

u ϭϩ The probabilities that 1, given s1 and s2 are

Pr~s1 ϭ H6u ϭϩ1!Pr~u ϭϩ1! Pr~u ϭϩ16s1ϭH!ϭ Pr~s1 ϭ H! p02 ϭ ϭ p. ~C3! p02 ϩ ~1 Ϫ p!02 1874 The Journal of Finance

~ ϭ ϭ ϭ ϭ ! When s2 confirms s1 either s1 H and s2 U or s1 L and s2 D , the player becomes overconfident and acts as if his precision were pC instead of p,so

Pr~s1 ϭ H,s2 ϭ U6u ϭϩ1!Pr~u ϭϩ1! Pr~u ϭϩ16s1ϭH,s2ϭU!ϭ Pr~s1 ϭ H,s2 ϭ U!

pC q ϭ . ~C4! pC ~2q Ϫ 1! ϩ ~1 Ϫ q!

~ Ͼ 0 ! When s2 is informative q 1 2 , this probability exceeds pC. When s2 does not confirm s1, the player does not become overconfident, so

Pr~s1 ϭ H,s2 ϭ D6u ϭϩ1!Pr~u ϭϩ1! Pr~u ϭϩ16s1ϭH,s2ϭD!ϭ Pr~s1 ϭ H,s2 ϭ D! p~1 Ϫ q! ϭ . ~C5! p~1 Ϫ q! ϩ q~1 Ϫ p!

~ Ͼ 0 ! When evaluated with an informative signal s2 q 1 2 , this probability is less than p. With a risk-neutral player, the price of the asset with value u can be calculated linearly using the preceding probabilities. The price at ~ ! u ϩ Ϫ time 0 P0 is, by definition, equal to 0. As can take on a value of 1or 1, the price is ~r!~ϩ1!ϩ~1Ϫr!~Ϫ 1! or, 2r Ϫ 1, where r is the probability that u is ϩ1.

6 ϭ ϭϪ 6 ϭ ϭ ~u ϭϩ 6 ϭ !Ϫ ϭ Ϫ ~ ! P1 s1 H P1 s1 L 2Pr 1 s1 H 1 2p 1 C6

6 ϭ ϭ ϭϪ 6 ϭ ϭ ϭ ~u ϭϩ 6 ϭ ϭ !Ϫ P2 s1 H,s2 U P2 s1 L,s2 D 2Pr 1 s1 H,s2 U 1

pC ϩqϪ1 ϭ ~C7! pC ~2qϪ1!ϩ~1Ϫq!

6 ϭ ϭ ϭϪ 6 ϭ ϭ ϭ ~u ϭϩ 6 ϭ ϭ !Ϫ P2 s1 H,s2 D P2 s1 L,s2 U 2Pr 1 s1 H,s2 D 1 pϪq ϭ . ~C8! pϩqϪ2pq

⌬ ϭ Ϫ ϭ ⌬ ϭ Ϫ @ # ϭ The price changes are P1 P1 P0 P1 and P2 P2 P1. E P1 0, so ~⌬ ⌬ ! ϭ @⌬ ⌬ # cov P1, P2 E P1 P2 . The probabilities of the eight possible outcomes are:

Pr~u ϭϩ1, s1 ϭ H, s2 ϭ U ! ϭ Pr~u ϭϪ1, s1 ϭ L, s2 ϭ D! ϭ pq02 ~C9! Investor Psychology and Market Reactions 1875

Pr~u ϭϪ1, s1 ϭ H, s2 ϭ U ! ϭ Pr~u ϭϩ1, s1 ϭ L, s2 ϭ D! ϭ ~1 Ϫ p!~1Ϫ q!02

~C10!

Pr~u ϭϩ1, s1 ϭ H, s2 ϭ D! ϭ Pr~u ϭϪ1, s1 ϭ L, s2 ϭ U ! ϭ p ~1 Ϫ q!02 ~C11!

Pr~u ϭϪ1, s1 ϭ H, s2 ϭ D! ϭ Pr~u ϭϩ1, s1 ϭ L, s2 ϭ U ! ϭ ~1 Ϫ p! q02. ~C12!

⌬ ⌬ The product P1 P2 can only take on two values, based on the various signal combinations:

[@⌬ ⌬ # ϭ ϭ ϭ@⌬ ⌬ # ϭ ϭ X P1 P2 s1 H,s2 U P1 P2 s1 L,s2 D

pCϩqϪ1 ϭ~2pϪ1!ͩ Ϫ~2pϪ1!ͪ. ~C13! pC~2qϪ1!ϩ~1Ϫq!

[@⌬ ⌬ # ϭ ϭ ϭ@⌬ ⌬ # ϭ ϭ Y P1 P2 s1 H,s2 D P1 P2 s1 L,s2 U pϪq ϭ~2pϪ1!ͩ Ϫ~2pϪ1!ͪ. ~C14! pϩqϪ2p

@⌬ ⌬ # ~ Ϫ ! ϩ ϭ ϩ Ϫ Combining, E P1 P2 can be written as 1 a X aY, where a p q 2pq. After some calculation, the two components of this expression become:

2~2p Ϫ 1!~2pq Ϫ p Ϫ q ϩ 1!~pC p ϩ pC q ϩ pq Ϫ 2pC pq Ϫ p! ~1 Ϫ a!X ϭ ~C15! pC ~2q Ϫ 1! ϩ ~1 Ϫ q!

aY ϭ 2p~2q Ϫ 1!~2p Ϫ1!~p Ϫ1!. ~C16!

Combining these two terms and a great deal of factoring produces the final result,

2q~2p Ϫ 1!~pC Ϫ p!~1Ϫ q! E @⌬P1 ⌬P2# ϭ Ͼ 0. ~C17! pC~2q Ϫ1! ϩ ~1Ϫ q!

~ ϭ ! When there is no overconfidence pC p this expression is zero and price changes are uncorrelated.

A Second Noisy Public Signal The model so far shows that overreaction can be exaggerated by a possible rise in confidence triggered by a noisy public signal. We now add a second noisy public signal to consider whether correction of mispricing is gradual. ' Signal s3 follows s2 and can take on values G or B. The precision of this signal is as follows:

Pr~s3' ϭ G6u ϭϩ1!ϭrϭPr~s3' ϭ B6u ϭϪ1!. ~C18! 1876 The Journal of Finance

This signal does not affect confidence. If the player becomes overconfident ~ ! and replaces p with pC after s2, then the player will continue to use pC as ' his measure of the precision of s1, regardless of whether s3 confirms s1.As there are two possible prices after the first signal and four possible prices after the second, there are eight possible prices after observation of the third signal. As above, by symmetry, only half of these prices need to be calculated. Using the conditional probabilities, the period three prices are:

pC qr Ϫ ~1 Ϫ pC !~1Ϫ q!~1Ϫ r! ' 6 ϭ ϭ ϭ ϭ ~ ! P3 s1 H,s2 U,s3' G ; C19 pC qr ϩ ~1 Ϫ pC !~1Ϫ q!~1Ϫ r!

pC q~1 Ϫ r! Ϫ ~1 Ϫ pC !~1Ϫ q!r ' 6 ϭ ϭ ϭ ϭ ~ ! P3 s1 H,s2 U,s3' B ; C20 pC q~1Ϫ r! ϩ ~1Ϫ pC!~1Ϫ q!r

p~1 Ϫ q!r Ϫ ~1 Ϫ p!q~1 Ϫ r! ' 6 ϭ ~ ! P3 s ϭH,s ϭD,s 'ϭG ; C21 1 2 3 p~1 Ϫ q!r ϩ ~1 Ϫ p!q~1 Ϫ r!

p~1 Ϫ q!~1Ϫ r! Ϫ ~1Ϫ p!qr ' 6 ϭ ~ ! P3 s ϭH,s ϭD,s 'ϭB . C22 1 2 3 p~1 Ϫ q!~1Ϫ r! ϩ ~1Ϫ p!qr

With two possible values for u, there are now sixteen possible sets of

$u,s1,s2,s3' % realizations. Only $s1,s2,s3' % are observed by the player, resulting in eight sets of possible signal realizations. When calculating the covariances of price changes, only half of these realizations can result in unique products of price changes, so we define

Aij [ ⌬Pi ⌬Pj6H,U,G ϭ⌬Pi⌬Pj6L,D,B; ~C23!

Bij [ ⌬Pi ⌬Pj6H,U, B ϭ⌬Pi⌬Pj6L,D,G; ~C24!

Cij [ ⌬Pi ⌬Pj6H, D,G ϭ⌬Pi⌬Pj6L,U,B; ~C25!

Dij [ ⌬Pi ⌬Pj6H, D, B ϭ⌬Pi⌬Pj6L,U,G. ~C26!

Each of these four possible products must then be weighted by its probability of occurrence in order to calculate the expected value of the products of the price changes ~the expected value of each price is zero!. The weights for the Aij component of covariance are:

Pr~H,U,G6u ϭϩ1!ϩPr~H,U,G6u ϭϪ1!ϭpqr02 ϩ ~1 Ϫ p!~1Ϫ q!~1Ϫ r!02,

~C27!

Pr~L, D, B6u ϭϪ1!ϩPr~L, D, B6u ϭϪ1!ϭpqr02 ϩ ~1 Ϫ p!~1Ϫ q!~1Ϫ r!02.

~C28! Investor Psychology and Market Reactions 1877

Proceeding in this manner, the covariances are:

E @⌬Pi ⌬Pj# ϭ @ pqr ϩ ~1 Ϫ p!~1Ϫ q!~1Ϫ r!# A ij

ϩ @ pq~1 Ϫ r! ϩ ~1 Ϫ p!~1Ϫ q!r#Bij

ϩ @ p~1 Ϫ q!r ϩ ~1 Ϫ p!q~1 Ϫ r!#Cij

ϩ @ p~1 Ϫ q!~1Ϫ r! ϩ ~1Ϫ p!qr#Dij . ~C29!

~ @⌬ ⌬ # ϭ ϭ ϭ ϭ Earlier calculations of E P1 P2 had A12 B12 X and C12 D12 Y, with the r and 1 Ϫ r factors from s3' summing to one.! To simplify the alge- bra, temporarily let all signals have the same precision ~i.e., q ϭ r ϭ p!, with pC replacing p as the perceived precision of the first signal if overconfidence occurs. Direct calculation of the covariances then shows that

2p~1 Ϫ p!~2p Ϫ1!~pC Ϫ p! E @⌬P1 ⌬P2#rϭqϭp ϭ Ͼ 0; ~C30! pC~2p Ϫ1! ϩ1 Ϫ p

3 2ppC ~ p Ϫ 1!~pC Ϫ p!~1Ϫ pC!~2p Ϫ1! E @⌬P ⌬P ' # ϭ ϭ ϭ Ͻ 0; ~C31! 1 3 r q p 2 @pC~2p Ϫ1! ϩ1 Ϫ p#@pC~2pϪ1!ϩ~1Ϫp! #

2 2 2 4p pC ~ p Ϫ 1! ~ pC Ϫ p!~pC Ϫ1!~2p Ϫ1! ~2pC Ϫ1! E @⌬P ⌬P ' # ϭ ϭ ϭ Ͻ 0. 2 3 r q p 2 2 @pC~2p Ϫ1! ϩ1 Ϫ p# @pC~2p Ϫ1! ϩ ~1Ϫ p! #

~C32!

@⌬ ⌬ # @⌬ ⌬ ' # By direct comparison, E P1 P2 rϭqϭp and E P2 P3 rϭqϭp are related by:

2p~1Ϫp!pC~1ϪpC!~2p Ϫ1!~2pC Ϫ1! E @⌬P ⌬P ' # ϭ ϭ ϭϪ 2 3 r q p 2 @pC~2p Ϫ1! ϩ ~1Ϫ p!# @ pC ~2p Ϫ 1! ϩ ~1 Ϫ p! #

ϫ E @⌬ P1 ⌬ P2 # rϭqϭp , ~C33! so the covariance between the date 2 and 3 price changes is negatively proportional to the covariance of the date 1 and 2 price changes. Consider the numerator N of the proportionality factor. The first three components, ~ Ϫ ! ϭ 0 ~ Ϫ 2p 1 p , are maximized when p 1 2; the next two components, pC 1 ! ϭ 0 pC , are maximized when pC 1 2. Because the last two components satisfy ~ Ϫ !~ Ϫ ! Ͻ Յ 0 2p 1 2pC 1 1, the N 1 8. In the denominator D, the expression ~ Ϫ ! ϩ ~ Ϫ ! ϭ ϭ 0 pC 2p 1 1 p is minimized when pC p 1 2, resulting in a mini- 0 ϭ mum of 1 2. The second component of D is similarly minimized when pC p ϭ 102, resulting in a minimum of 104. So D Ն 108. Because N Յ 108, the ratio N0D Յ 1. Therefore, the negative covariance between date 2 and date 3 price changes must be, in absolute value, less than or equal to the positive 1878 The Journal of Finance covariance between period one and period two price changes, resulting in an overall one-period covariance that is positive. When q ϭ r differs from p, direct calculation of covariances shows:

E @⌬P1 ⌬P2#rϭq Ͼ 0; ~C34!

E @⌬P1 ⌬P3' #rϭq Ͻ 0; ~C35!

E @⌬P2 ⌬P3' #rϭq Ͻ 0. ~C36!

Now let the signal s3' have a precision of r that differs from both precisions of p and q. Proceeding as above, the covariances satisfy

E @⌬P1 ⌬P2# Ͼ 0; ~C37!

E @⌬P1 ⌬P3' # Ͻ 0; ~C38!

E @⌬P2 ⌬P3' # Ͻ 0; ~C39!

E @⌬P3' ⌬P3# Ͼ 0; ~C40!

E @⌬P1 ⌬P3# Ͻ 0, ~C41!

The magnitude of E @⌬P2 ⌬P3' # varies nonmonotonically with q.Asrrises ~the precision of s3' is increased!, direct calculation shows that E @⌬P2 ⌬P3' # becomes more negative ~increases in absolute value!.Asrr0.5, this covariance approaches zero. Thus, when the second noisy public signal is not very informative, this negative single-lag covariance becomes arbitrarily small in absolute value.

Confidence increases when s2 confirms s1, but its effects are mitigated as s2 becomes more informative. Thus, an increase in the precision of s2 has an ambiguous effect on E @⌬P2 ⌬P3' #. This increase results in a greater likelihood of overconfidence occurring, yet also places greater, rational, confi- ~ dence in s2 itself, yielding less leverage to the effects of overconfidence. At the extreme, a value of q equal to one yields the greatest chances of s2 confirming s1, yet results in zero values for all covariances as the perfect ! information of s2 entirely determines all subsequent prices. Based on simulation, it appears that the greater information resulting from higher values of q tends to overshadow the increased likelihood of overconfidence, resulting in generally lower absolute values for E @⌬P2 ⌬P3' #. Larger values of r, the precision of s3' , result in more negative values of E @⌬P2 ⌬P3' #. In this case, a more informative second noisy public signal can only place less weight on previous signals and result in a stronger correction of the previous overreaction. Thus, the final one-period covariance is more negative as the precision of s3' rises. Investor Psychology and Market Reactions 1879

Appendix D: Covariance Calculations for the Dynamic Model (Section III.A.1) Since the probability p is an exogenous constant, and the probability that the date 1 price move was positive is 102, by the law of iterated expectations

~ Ϫ Ϫ ! ϭ @ @$~ Ϫ !~ Ϫ !%6 ## cov P2 P1, P1 P0 Es2 E P2 P1 P1 P0 s2

4 2 2 ksu ~su ϩ se ! ϭ Ͼ 0; ~D1! 2 2 2 2 2 2~su ϩ sC ! ~su ϩ sC Ϫ k!

~ Ϫ Ϫ ! ϭ @ $$~ Ϫ !~ Ϫ !%6 %# cov P3 P2, P2 P1 Es2 E P3 P2 P2 P1 s2

2 2 2 2 2 su @k su ϩk~se Ϫ sC !# ϭϪ Ͻ 0. ~D2! 2 2 2 2 2 2~su ϩ sC !~su ϩ sC Ϫ k!

Further,

4 2 2 su ~se Ϫ sC ! cov~P Ϫ P , P Ϫ P ! ϭ E @~u Ϫ P !P # ϭϪ Ͻ 0. ~D3! 3 1 1 0 1 1 2 2 2 ~su ϩ sC !

Direct calculation shows that

cov~P3 Ϫ P3' , P3' Ϫ P2!

6 2 2 2 su s ~se Ϫ s ! ϭ ͫ C C 2 2 2 2 2 2 2 2 2 ~su ϩ sC ! @su ~sC ϩ sp ! ϩ sC sp #

2 2 2 ~sC Ϫ k!~se ϩ k Ϫ sC ! ϩ ͬϾ 0; ~D4! 2 2 2 2 2 2 2 2 ~su ϩ sC Ϫ k! @su ~sC Ϫ k ϩ sp ! ϩ ~sC Ϫ k!sp #

cov~P3 Ϫ P, P2 Ϫ P1!

2 4 2 2 ksp su~kϩse ϪsC! ϭϪ Ͻ 0; ~D5! 2 2 2 2 2 2 2 2~sC Ϫ k ϩ su !@~sC Ϫ k!~su ϩ sp ! ϩ sp su #

' cov~P3 Ϫ P3 , P1 Ϫ P0!

4 2 2 2 1 su s ~se Ϫ s ! ϭϪ ͫ p C 2 2 2 2 2 2 2 2 ~su ϩ sC !@sC~su ϩ sp ! ϩ su sp #

4 2 2 2 su sp ~k ϩ se Ϫ sC ! ϩ ͬ Ͻ 0; ~D6! 2 2 2 2 2 2 2 ~su ϩ sC !@~sC Ϫ k!~su ϩ sp ! ϩ sp su # 1880 The Journal of Finance

' cov~P3 Ϫ P2, P1 Ϫ P0! 6 2 2 2 1 su s ~se Ϫ s ! ϭϪ ͫ C C 2 2 2 2 2 2 2 2 2 ~su ϩ sp ! @sC~su ϩ sp ! ϩ su sp # 6 2 2 2 su ~sC Ϫ k!~k ϩ se Ϫ sC ! ϩ ͬϽ 0; ~D7! 2 2 2 2 2 2 2 ~su ϩ sC Ϫ k!@~sC Ϫ k!~su ϩ sp ! ϩ sp su # and that

cov~P3' Ϫ P2, P2 Ϫ P1! 6 2 2 2 ksu~sC Ϫ k!~se ϩ k Ϫ sC ! ϭϪ Ͻ 0. ~D8! 2 2 2 2 2 2 2 2 2~sC ϩ su Ϫ k! @sp su ϩ ~sp ϩ su !~sC Ϫ k!#

The three single-lag contiguous covariances are given by equations ~D1!, ~D4!, and ~D8!. Comparing the covariances ~D1! and ~D8!, it is evident that 2 the sum of these two covariances will be negative so long as sp is sufficiently large because the latter ~negative! covariance varies inversely with 2 2 sp and the former covariance does not depend on sp . The covariance in ~D4! is always positive, so unconditional momentum ~defined as the simple arith- 2 metic average of the three single covariances! obtains if sp is sufficiently small. As k r 0, the covariance in equation ~D8! goes to zero, and the covariance in equation ~D4! remains strictly positive. Thus, if k is sufficiently small, the sum of these two covariances will be positive. Because equation ~D1! is always nonnegative, unconditional momentum will obtain if k is sufficiently small.

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